on the transition of the cylinder wake - bernd noack

On the transition of the cylinder wakeHong-Quan Zhang,a)b) Uwe Fey, and Bernd R. Noack,a)Afax-Planck-Institutfiir Stromungsforschung, Bunsenstrasse 10, D-37073 Gottingen, Germany

Michael K6nig and Helmut EckelmannInstitut ffir Angenandte Mechanik und Strimungsphysik der Universitat, Bunsenstrasse 10,D-37073 Gittingen, Germany

(Received 9 March 1994; accepted 8 December 1994)

The transition of the cylinder wake is investigated experimentally in a water channel and iscomputed numerically using a finite-difference scheme. Four physically different instabilities areobserved: a local "vortex-adhesion mode," and three near-wake instabilities, which are associatedwith three different spanwise wavelengths of approximately 1, 2, and 4 diam. All four instabilityprocesses can originate in a narrow Reynolds-number interval between 160 and 230, and may giverise to different transition scenarios. Thus, Williamson's [Phys. Fluids 31, 3165 (1988)]experimental observation of a hard transition is for the first time numerically reproduced, and isfound to be induced by the vortex-adhesion mode. Without vortex- adhesion, a soft onset ofthree-dimensionality is numerically and experimentally obtained. A control-wire technique isproposed, which suppresses transition up to a Reynolds number of 230. © 1995 American Instituteof Physics.

1. INTRODUCTION

The incompressible flow around a circular cylinder rep-resents one of the most investigated prototypes of bluff-bodywakes. The properties of this flow depend on the Reynoldsnumber Re= UDI r, where U is the velocity of the oncomingflow, D is the diameter of the cylinder, and v is the kinematicviscosity of the fluid. Most authors agree that the transitionscenario contains a two-dimensional (2-D) instability from a2-D steady to a 2-D periodic wake at Re, -45 and a three-dimensional (3-D) transition at a Reynolds number Re, 2 be-tween 150 and 210. While the onset of periodicity has beenconclusively identified as a supercritical Hopf bifurcation, 1-3

there still exist many contradictory results regarding the on-set of three-dimensionality. Williamson4 ' 5 experimentally ob-serves a hard hysteretical transition towards an irregularwake, accompanied with a jump of the Strouhal number St-fD/U (f: dominant frequency) and the base pressure6 atthe transition Reynolds number 180. The jump of the Strou-hal number is also confirmed for different cylinder end con-ditions by K6nig, Noack, and Eckelmann7 and by Bredeet al.8 In contrast, seemingly all recent 3-D numerical simu-lations of Karniadakis and Triantafyllou,9 Tomboulides, Tri-antafyllou, and Karniadakis,10 and Noack and Eckelmann""1, 2

indicate a soft transition to a 3-D, periodic flow.Recent values for the critical Reynolds number Re, 2

range from 150 to 210 in the cited references. Most authorsreport that the 3-D wake at Re-Re, 2 is characterized by awake pattern with a dominant spanwise wavelength, Az c2-

The reported values for this wavelength lie between 1-4diam.13-16 While Williamson observes a transition from largespanwise patterns with a wavelength X,- 3D (his A mode) toa fine-scale pattern with a siz&'of ID (his B mode) near

')Also at the Deutsche Forschungsanstalt fur Luft- und Raumfahrt, Institutfur Strumungsmechanik, Bunsenstrasse 10, D-37073 G6ttingen, Germany.

b)Permanent address: Department of Mechanics, Tianjin University, Tianjin300072, People's Republic of China.

Re-230, the global stability analysis of Noack and Eckel-mann yields an intermediate value of 1.8D. Gerrard'7 em-phasizes that also wavelengths of ten and more diameters canbe found in some experimental works. Yet, these large struc-tures seem to be induced by the strong perturbations at theexperimental end conditions and have not been reproducedin more recent works with improved experimental facilities.

In addition to the wake patterns with a dominant span-wise wavelength, also localized vortex deformations,Williamson's' 4 "vortex dislocations," are experimentally ob-served at transitional Reynolds numbers. At these "disloca-tions," the von Karmin vortices seem to "adhere" at asteady or slowly moving point at the cylinder for many pe-riods. Hence, the vortex-adhesion mode may be a more suit-able term for this phenomenon. Since there seems to exist nopublished numerical simulations of the vortex-adhesionmode, it cannot be conclusively settled whether these struc-tures are only end effects due to the finite aspect ratio or ifvortex adhesion may also be a self-sustaining shedding statefor an infinitely long cylinder. While the vortex-adhesionphenomenon has been shown to have a large effect on the farwake,14 their influence on the onset of three-dimensionalityin the near wake is not well investigated so far. A furthercomplication of the cylinder wake transition is the experi-mental observation that the far-wake structures and dominat-ing time scales are less organized and significantly largerthan the near-wake features.18"19

In the present publication, the cylinder wake transition isinvestigated in order to elucidate the reasons for the discrep-ancies in the literature. For this purpose, an accurate 3-Dfinite-difference scheme was developed2 0 and a water chan-nel was constructed. In Sec. II, the construction and vali-dation of the employed Navier-Stokes solver are described.In Sec. III, the experimental setup is outlined. The numericaland experimental results on the cylinder wake transition are

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cide with coordinate lines (=const or cost. The dimen-sions of the ( and 27 coordinates are chosen so that the par-tition of the grid consists of unit squares, i.e., the ( and 77coordinates of the grid lines are integers.

Following Thompson et aL,22 the mapping x=x(, y7),y =y(6, 77) from the computational to the physical domain isobtained by numerically solving the following system ofPoisson equations with Dirichlet boundary conditions:

d2x t 2x 2 x ,, ax ax \v + yg d=1 _iv P +Q---

e d~ T7_ IaT a1 '7

1111

'6*

T.7*

-I -I- r .8liii

FIG. 1. Physical (top) and computational (bottom) domain for the finite-difference scheme.

detailed in Sec. IV. Finally (Sec. V), the main findings aresummarized and discussed.

II. NUMERICAL METHOD

In this section, the incompressible Navier-Stokes solveris described. This solver is based on a finite-differencemethod for generalized coordinates on a boundary-fittedgrid.22 First (Sec. II A), the generation of the grid is outlined.In Sec. II B, the finite-difference scheme and the employedboundary conditions are discussed. Finally (Sec. II C), thenumerical scheme is validated for 2-D and 3-D solutions.

A. Grid generation

In the following, the flow is described in a Cartesiancoordinate system x,y,z, where the x axis is aligned with theoncoming flow, the z axis coincides with the axis of symme-try of the cylinder, and the y axis is perpendicular to the xand z directions. These coordinates are assumed to be non-dimensionalized with the cylinder diam D, i.e., the cylindersurface is described by Jx 2+yy = 112.

The numerically resolved physical domain in the x,yplane consists of a half-ellipse in the front joined by a rect-angle for the wake region (see Fig. 1, top). This domain issimilar to the one used by Karniadakis and Triantafyllou23'9

for 2-D and 3-D cylinder wake simulations, except that theyuse a half-circle in the upstream region. Following the workof Thompson et aL, the physical domain is mapped on asimpler T-shaped computational domain (see Fig. 1, bottom).The cylinder is mapped onto the line F*, the outflow bound-ary onto r* , the upper and lower boundaries of the wakerectangle onto r4 and F*, respectively, and the upstreamhalf-ellipse onto F* and F*. The boundaries F* and F*denote the cut y = 0 in the front region. The computationaldomain is described by an orthogonal coordinate system A, y.In this system, all boundaries of the physical domain coin-

d2 y 82y d2 y 2 8y 8y

a 2, d+ d7 _i P deQ d71

where a=t92x/d712 + d2y/da 2, /3=(dx/d)(dx/1a'1) +(y/adj(ayldza), y=da2 x/a 2 +d 2 ylda 2 , and J= d(x,y)/d(6, 77)=(ax/da)(dy/daT)-(dx/daq)(dy/da). With the source func-tions P(6, y) and Q({,77), the density of grid lines in thephysical domain can be increased or decreased in regionswith large or small gradients (see, e.g., Chap. 13 ofFletcher2 4). In the present publication, P = O andQ=a1 exp(-ccq), where coefficients at and cl are chosen inorder to increase the radial resolution close to the cylindersurface. In the wake region, the source terms vanish and themapping ((; 77)-(x;y) becomes locally orthonormal.

The size of a 2-D computational domain around the cyl-inder is described by three parameters; the size of the up-stream region Xi, the x coordinate of the outflow boundaryX0 , and the width of the wake rectangle Y, The inflowboundary is a half-ellipse with a principal axis ratio of 1:2.The numerically resolved physical domain is thereforebounded by -Xi<x<Xo and Iyl<Yf 2 (see Fig. 2, bot-tom). The grid C, illustrated in Fig. 2 (bottom), is employedfor the numerical computations. For validation purposes, twoother grids with either smaller dimensions (grid A) or largergrid spacings (grid B) are generated (see Fig. 2, top andmiddle). The geometric parameters and the resolution of thegrids are displayed in Table I. Here, N6 and N. represent thenumber of grid points in the 6 and y7 directions. Here Axoand AyO are the grid spacings in the x and y directions,respectively, near the outflow boundary. Also, As, denotesthe grid spacing on the cylinder surface in the normal direc-tion.

Since the computation is effected in the computationaldomain, the Navier-Stokes equations have to be expressedin terms of the coordinates (,x7. The transformation of thespatial derivatives in the physical domain read as

a 1 a d a \a x J J 7 'f a dJ

d 1 7 a d detc.

dy J t X ,+fd71/e

Further details can be inferred from most textbooks of com-putational fluid dynamics, for instance, from Chap. 13 ofFletcher.2 4

780 Phys. Fluids, Vol. 7, No. 4, April 1995

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I 1-I'l. . . 1 .1 . IT

T, .n

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Zhang et al.


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FIG. 2. Grids A, B, and C (top to bottom).

B. Finite-difference scheme

The velocity field in terms of the location x=(x;y;z)and the time t is described by u=(u;v;w), where u, v, andw are the components in x, y, and z directions. The pressureis denoted by p. In the following, all independent and depen-dent variables are assumed to be nondimensionalized withthe diam D and the velocity of the oncoming flow U. Theevolution of the incompressible velocity field is described bythe Navier-Stokes and the continuity equations,

du l~~Vudu + (u-V)u= -VP+ 1 V'U,

V *U=0.

For the numerical computations, the boundary condi-tions cannot be imposed at infinity, but have to be specifiedat the boundary of the numerically resolved physical domain.At the cylinder surface, the no-slip condition u=0 is en-forced. The normal pressure gradient is set equal to zero, i.e.,a homogeneous von Neumann condition is imposed on p.The exact condition may be derived from the Navier-Stokes

TABLE I. Parameters of the grids used for the 2-D validation.

AxOGrid Xf X0 Y, NfXN, =AY0 As, a, cl

A 3.0 11.5 6.0 96X 96 0.1667 0.010 1000.0 0.3B 6.0 16.0 12.0 102X 90 0.2500 0.012 1000.0 0.3C 6.0 16.0 12.0 144X 144 0.1667 0.008 1000.0 0.2

equations. Yet, this condition converges to the von Neumanncondition in the boundary-layer approximation at large Rey-nolds numbers and is numerically found to yield insignifi-cantly different solutions. At the inflow boundary, the veloc-ity and the pressure are assumed to be uniform, u=(U;0;0)and p =const. At the sides of the wake rectangle y = + YwI2,vanishing normal gradients for all flow variables are as-sumed, i.e., Ou/dy = dpady = 0. The outflow boundary condi-tion at x=X0 should allow the vortices to leave withoutcausing upstream perturbations. This is achieved by requir-ing vanishing third-order x derivatives, i.e.,Auidx3 =a3 p/dx3 =0. In the spanwise direction, the flow isassumed to be periodic with wavelength L.

The initial condition for a 2-D computation (u;v;w)=(1;0;0) corresponds to an impulsive start. Asymmetric2-D solutions may be obtained by adding a small perturba-tion to the uniform initial condition. Post-transient 2-D solu-tions superimposed by a 3-D perturbation are typically usedas initial conditions for the 3-D computation. In a Reynoldsnumber range of 160-230, small 3-D perturbations result ina smooth onset of three-dimensionality, whereupon large lo-calized 3-D perturbations give rise to a hard hysteretical tran-sition for a sufficiently large spanwise domain size L (seeSec. IV).

For 2-D computations, the velocity u at the node ((; 7)

= (i;j) and the time level n is denoted by ups. The velocity isexpanded as a Taylor series in terms of the time,

at t^ m atm am=M! latm

m=1 II

The temporal order of a finite-difference method depends onthe number of the considered terms in the truncated expan-sion. Methods with a temporal order larger than unity containhigher-order spatial derivatives that are not present in theoriginal Navier-Stokes equations.25 The implementation ofthese additional terms may make the scheme less economi-cal, and also less accurate if the new terms are not treatedproperly. The time step At depends on the grid spacing,which must be small enough for the resolution of the veloc-ity gradients, particularly near the cylinder surface. In thepresent publication, the chosen time step At varies from10-2 to 10-3. If only the first term in the temporal expansionis taken, the truncation error is insignificant,

Un+j 1=ug +At- + O (A t

2 ).if If a9t

For the temporal integration, a MAC-type finite-difference scheme (see, for instance, Chap. 17 in Fletcher 24 )is employed. In this scheme, the iteration is carried out intwo steps on a staggered grid, where the pressure nodes andthe velocity nodes are displaced by half a grid spacing. In thefirst step an intermediate velocity u* is computed from theflow variables at time level n, according to

,=un 2-1 1 V .=U i+ AtI (U.V)u+-VuiU~~LJ ~ Re j.()

Phys. Fluids, Vol. 7, No. 4, April 1995 Zhang et a/. 781


TABLE II. Strouhal number St, mean drag coefficient CD, and RMS am-plitude of the lift coefficient Cl at Re=100 for 2-D grids A, B, and C.

Grid St CD CL

A 0.190 1.548 0.225B 0.172 1.421 0.247C 0.173 1.425 0.250

The RHS of Eq. (1) represents the discretization of the spa-tial derivatives at the node (i;j) and the time level n. In thesecond step, the velocity is calculated for time level n + 1:

unj1 =u 1- At Vpl-. (2)

Before the second step (2), the Poisson equation for pressure

Pi

V2P I*= V u*At

is solved. While the intermediate field u* is generally notsolenoidal, the velocity field at time level n + 1 can be shownto fulfill the discretized incompressibility condition.

The first-order spatial derivatives in the convective termof the Navier-Stokes equations are upwind discretized usingfour grid points (for details see Ref. 20). The second-orderderivatives in the diffusion term are approximated by three-point central differences. Thus, a third-order upwind schemeis obtained, which is known to prevent nonphysical oscilla-tion and which is expected to have negligible numerical vis-cosity on the employed boundary-fitted grid.

C. Validation

In order to study the effect of the domain size and thegrid resolution on the 2-D solutions for the cylinder wake,computations are performed at Re=100 on three differentgrids (A, B, C). At this Reynolds number, 2-D parallel vortexshedding can be achieved in the laboratory with a carefultreatment of the end conditions (Eisenlohr and Eckelmann 2 6).Therefore, the influence of domain size and grid resolutioncan be assessed by investigating the discrepancy between theexperimental and numerical Strouhal number values. Thecomputed Strouhal numbers, mean drag coefficients, andRMS amplitudes of the lift coefficients are enumerated inTable II. The experimental Strouhal number at Re=100 is0.167 using the empirical formula of K6nig, Noack, andEckelmann.7 From Table HI the numerical values for theStrouhal numbers are seen to be larger than the experimentalvalue. The discrepancy is around 3% for grids B and C andabout 12% for the smaller grid A. The improved grid reso-lution of grid C-as compared to grid B-seems to havelittle effect on the solution. In contrast, the numerical data forthe Strouhal number appear to converge rapidly to the ex-perimental value with increasing domain size. A similar ten-dency is reported by Karniadakis and Triantafyllou 9 for theircylinder wake simulation with a spectral method.

The performance of the finite-difference method in termsof the Reynolds numbers is investigated for the 2-D wake inthe range 40<Re<300. For this computation grid C is em-

0.1

0.1

0.1

0.11

C,1.1

1.:

0

C

0.

0.1

0.-

0.:

50 100 150 200 Re 300

FIG. 3. Strouhal number St (top), mean drag coefficient CD (middle), andRMS amplitude of the lift coefficient C' (bottom) in terms of the Reynoldsnumbers Re. The solid circles (0) denote our numerical values. The solid Stvs Re curve represents the empirical formula of Konig et aL7 for 2-D shed-ding. The symbols (+) and (X) refer to CD measurements of Tritton37 andWieselsberger, 3 8 respectively.

ployed. In Fig. 3 (top), the solid circles represent the com-puted Strouhal numbers. The curve is based on the empiricalformula of Konig et al.7 In the whole Reynolds number in-terval the computed values are about 3% larger than the ex-perimental results. For an increased size of the computationaldomain, a smaller discrepancy is expected.

The mean drag coefficient CD and the RMS amplitude ofthe lift coefficient C' are presented in Fig. 3 (middle andbottom) in terms of the Reynolds number. The mean dragcoefficients obtained in the 2-D computation are up to 20%larger at Re=300 than the experimental ones, since in thelaboratory the flow is superimposed by 3-D fluctuations afterthe transition. For the lift coefficients, no experimental datahas been found at low Reynolds numbers.

In the present work, also the influence of different out-flow boundary conditions on the numerical results is inves-tigated. The flow state is not noticeably changed, when thefirst, second, or the third downstream derivatives of flowvariables are set equal to zero, except for minor differencesnear the outflow boundary.

In 3-D computations the numerical solution is affectedby the spanwise domain size L and the spanwise grid spacingAz-in addition to the 2-D grid. Therefore, the performanceof the finite-difference scheme is studied for the different


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Zhang et al.


TABLE m. Parameters of the grids used for the 3-D validation.

No. Grid Az L

1 A 0.1 32 B 0.1 33 C 0.1 34 C 0.1 65 C 0.1 96 C 0.05 37 C 0.2 3

2-D grids A, B, and C and various spanwise resolutions anddomain sizes. In Table III, the parameters of the employed3-D grids are listed. The first three entries #1, #2, and #3 inTable III correspond to grids A, B, and C with the samespanwise spacing Az= 0. iD and the same spanwise dimen-sion L = 3D. In the 3-D grids, #3, #4, and #5, the spanwisewavelength is varied: L =3D, 6D, and 9D, using grid C andthe same spanwise resolution Az= 0. ID. In the cases #3, #6,and #7, the spanwise resolutions are Az= 0.1D, 0.05D, and0.2D, respectively, while the 2-D grid and spanwise dimen-sion remain the same.

Table IV lists the corresponding numerical results for thedeveloped 3-D solution, including Strouhal numbers, meandrag coefficients, RMS amplitudes of the lift coefficients,and RMS and maximum values of the spanwise velocity waveraged along a line parallel to the z axis for x/D = 1.5 andy=O. The values for wRms and wm.u: can be viewed as am-plitudes of three-dimensionality, since they vanish for 2-Dflow. The Reynolds number chosen for this validation is 300,at which experiments show that the 3-D wake is dominatedby fine-scale structures with a spanwise wavelength of about1D.27 Using 3-D grid #1 with the smallest 2-D domain, theSt and CD values are significantly larger than for the othergrids (see Table IV). For grid #2 with the coarse discretiza-tion of the x,y plane and for grid #7 with a coarse spanwiseresolution, WRMS and wmax are significantly smaller than thatof finer grids, indicating that the 3-D instability processesseem to be insufficiently resolved.

Therefore, we use grid C as the x,y projection for the3-D computations. Thus, reasonably accurate 2-D solutionsand a good resolution of the 3-D fluctuations in the x,y planeare guaranteed. A spanwise grid spacing of Az= 0. iD ap-

TABLE LV. Strouhal number St, mean drag coefficient CD, RMS amplitudeof the lift coefficient C,, RMS and maximum value of the spanwise veloc-ity component, wpm5 and wm,, at Re=300 for the 3-D grids, listed in Table111. The statistics of the spanwise velocity component are spatiotemporalaverages with respect to the time and the z coordinate for x/D = 1.5 andy=

0.

No. St CD C. WRM5 Wm,

1 0.232 1.432 0.4217 0.0828 0.33762 0.210 1.292 0.4470 0.0667 0.28273 0.212 1.311 0.4507 0.0854 0.36374 0.212 1.278 0.4387 0.0886 0.37255 0.212 1.260 0.4298 0.0895 0.37366 0.212 1.308 0.4513 0.0901 0.37647 0.212 1.312 0.4506 0.0683 0.2984

U L>~

w~~~y

FIG. 4. Principal sketch of the test section. The cylinder C is mountedvertically. The hydrogen-bubble wire W is mounted at the end plates E.

pears to be fine enough to resolve the small secondary vortexstructures with spanwise wavelengths around 1D. For 3-Dstructures of much larger wavelength, a slightly larger span-wise spacing of 0. 15D is used for economical reasons.

Ill. EXPERIMENTAL SETUP

The experiments have been carried out in a water chan-nel. A hydrogen-bubble method is applied to obtain a visualimage of the cylinder wake. The channel is especially opti-mized for investigations at free-stream velocities between 3and 15 cm/s. The honeycomb and various screens are modi-fied so that a stationary, uniform velocity profile is achievedin the test section, which is 250 mm wide and 330 mm high.A detailed description of the channel is given by Fey.21 Here,only the main features will be summarized.

The visualizations of the cylinder wake are carried out inthe test section just behind the 4:1 contraction of the nozzle(Fig. 4). The cylinders have a polished surface and are madefrom stainless steel. Their diameters are 2, 3, and 4 mm, thecorresponding aspect ratios and end conditions are listed inTable V. Thus, the transition range (180<Re<300) is cov-ered by the above velocity range, for which the channel isoptimized. The cylinders are mounted vertically in the testsection and are bounded by end plates or end cylinders inorder to minimize end effects.

A 25 ,um diam platinum-iridium wire is located at dif-ferent positions in the test section and serves as thehydrogen-bubble wire. This wire is fixed directly at the con-fining end plates or at the end cylinders (see Fig. 4). Byrotating the cylinder around its axis, the bubble wire can belocated in different angular positions with respect to the front

TABLE V. Employed experimental setups.

No. D (mm) LID End conditions

1 2 133 End plates2 3 50 End cylinders3 3 93 End plates4 4 71 End plates

Phys. Fluids, Vol. 7, No. 4, April 1995 Zhang et al. 783


stagnation point of the cylinder. Thus, the hydrogen bubblescan be introduced in different sheets of the separating shearlayers. In the transition range, the location of the hydrogen-bubble wire proves to be an important parameter for thestructures that can be observed in the cylinder wake. In somelocations, the visualization wire can have the effect of a con-trol wire, the presence of which may drastically change thewake features. In another configuration, the effect of the vi-sualization wire on the wake features is avoided by placing it15 cylinder diameters upstream and slightly offset from theplane y = 0. In the present investigation, this configuration isemployed for reasons of comparison. This aspect is dis-cussed in detail in the following sections.

The Reynolds number of the hydrogen-bubble wire is ofthe order of unity in the present experiments. Hence, thiswire cannot give rise to vortex shedding. A hydroelastic cou-pling of this wire with the vortex street behind the cylindercannot a priori be excluded and is carefully examined. Typi-cally, vibrations are easily observed by eyesight, since thecharacteristic frequency of the vortex shedding is only a fewHertz. For a more precise detection of possible wire vibra-tions, a nearly unidirectional light source is used. This lightsource is directed on the visualization wire at a small angleand a photodiode is located in the light ray reflected from thewire. Thus, small vibrations of the wire are indicated by thephotodiode device. Hardly noticeable oscillations of the wireare only detected for a position near (x,/D;y,/D)'(1,0).This setup corresponds to the retarded onset of three-dimensionality described in Sec. IVE. The wire remainssteady for all other investigated positions in the whole regu-lar and transitional Reynolds number range.

Images of the wake flow in the x,z plane are recorded bya CCD camera connected to a videotape recorder and a digi-tal image processing system controlled by an IBM-compatible PC. Before each experiment, the velocity in thetest section is determined optically. A hydrogen-bubble wireis fixed between the upper and lower tunnel walls at x= -60mm, y =-66 mm and is pulsed to generate vertical timelines in the test section. At this location, the wire is inside thecore region of the test section and has no noticeable influ-ence on the cylinder wake. The velocity is determined fromthe time interval between two pulses and the distance of thecorresponding time lines. This method proved to be of highaccuracy and reproducibility.

The characteristic spanwise wavelength is measuredmanually from digitized images of the CCD camera. First,the x,z view section of the camera is determined by refer-ence marks. Then, the camera records several hundred shed-ding periods. After an evaluation of the video, selected pic-tures are digitized and employed for measuring the length ofthe spanwise structures.

IV. PHENOMENOLOGY OF THE CYLINDER WAKE

In this section, numerical and experimental evidence forfour distinct 3-D instability processes, which give rise todifferent transition scenarios, are presented. In Secs. IV Aand IV B, Williamson's 4 vortex dislocations-our vortex-adhesion mode-and Williamson's 27 A and B mode withspanwise wavelengths of 4 and 1 diameters, respectively, are

experimentally reproduced and for the first time numericallysimulated. In Sec. IV C, the 3-D Floquet mode predicted byNoack, K6nig, and Eckelmann1 5 with a low-dimensionalGalerkin method is identified as a separate 3-D instability.This instability has a spanwise wavelength of roughly twodiameters and is called the C mode in the following. Thus,the three global shedding states, the A, B, and C mode, caneasily be distinguished in terms of their characteristic span-wise wavelengths. A detailed comparison of their spatialstructures is presented in Sec. IV D. In Sec. IV E, a control-wire technique for the suppression of three-dimensionalityup to a Reynolds numbers of approximately 230 is presented.Finally (Sec. IV F), the interaction of the four instability pro-cesses is described.

A. Vortex-adhesion mode

The finite-difference computations can reproduceWilliamsonii 4 spot-like "vortex dislocations" for a suffi-ciently large domain size L (see Fig. 5). In this figure, thewavelength of the spanwise domain is L/D=24, which isone order of magnitude larger than in previoussimulations 2 The isopressure surface from the numericaldata (Fig. 5, top) looks similar to the experimental hydrogen-bubble-wire visualizations (Fig. 5, bottom). Both structuresidentify essentially the primary von Kirmin vortices. Thesevortices seem to "adhere" to slowly migrating points on thecylinder surface. Hence, we propose the term vortex-adhesion mode for this phenomenon. This mode is self-sustaining in the range 160<Re<230.

In the experiments, vortex-adhesion points originate atthe ends of the cylinder as the Reynolds number is slightlyincreased above the critical value 160. At supercritical valuesRe>160, these points occur intermittently along the wholecylinder span with a nearly uniform statistical distribution(see Fig. 6, bottom). The amount of adhesion points tends toincrease with the Reynolds number. At Re<180, the vonKremlin vortices typically shed obliquely between twoneighboring adhesion points. At Re-'180, many regions be-tween two neighboring adhesion points are often character-ized by A-mode patterns with a few spanwise periods (seeSec. IV B). If the B mode (see Sec. IV B) dominates thevortex shedding at Re>230, no pronounced adhesion pointsare observed. As the Reynolds number is slowly decreasedbelow Re= 160, the adhesion mode is finally replaced byparallel or oblique shedding. Under suitable conditions, aself-sustaining adhesion mode is also experimentally ob-tained in the range 140<Re<160, for instance, when theReynolds number is decreased sufficiently rapidly from ir-regular values, say Re=800, to a subcritical value. Then, theshedding is characterized by one or a few adhesion points,which finally assume steady asymptotic positions, which areconstant for the whole period of investigation, i.e., manythousand shedding periods (see Fig. 6, top).

The process of the creation and the propagation of theadhesion points suggests that the adhesion mode affects thevortex shedding in the range 160<Re<230 for arbitrarilylarge aspect ratios and any end condition in the post-transientstate. In the present experiments, the vortex-adhesion modeis observed for four different setups with aspect ratios from

784 Phys. Fluids, Vol. 7, No. 4, April 1995 Zhang et al.


I 'I

1(1

FIG. 6. Experimental flow visualization of the vortex-adhesion mode at asubcritical Reynolds number Re=152 (top) and at a supercritical value Re=176 (bottom). Each figure displays the cylinder (left), the two end plates(see the top and bottom side of the frame), and the hydrogen-bubble wirebeing fixed at the end plates (the beginning of a streak surface). The aspectratio is given by LID=133 and the wire is located at(x,1D;y,1D)=(l6;2).

FIG. 5. Illustration of the vortex-adhesion mode from the numerical com-putation (top) and from the experimental flow visualization (bottom). Theview section is given by -0.5<x/D<16, O<z/D<24. In both illustra-tions, the cylinder is situated at the left and the flow direction is from left toright. The numerically obtained vortex-adhesion mode (top) is described byan instantaneous isopressure surface p= -0.2 at Re= 160. In the experi-ments (bottom), the wake structures at Re=161 are visualized with ahydrogen-bubble wire positioned at (xfD;y,/D)=(l.5;0). Thus, the hy-drogen bubbles (dark regions) concentrate on both sides of the von Kxrminvortex street. The aspect ratio of the cylinder is given by L/D = 93. Similarstructures are also obtained with a wire located far upstream xID-50 andfar downstream (see Fig. 6).

50 to 133 and two kinds of end conditions (see Table V).Naturally, the transient time in which the adhesion pointsreach the midspan region increases with the aspect ratio. Thissituation is analogous to the transient "phase fronts" be-tween parallel and oblique shedding in the regular Reynoldsnumber range 50<Re<160. These phase fronts originate atthe cylinder ends and move toward the midspan region withconstant speed. 5,28 Finally, the whole span of the cylinder isgoverned by a chevron pattern or oblique shedding. Thus,even laminar shedding seems to be always affected by theend conditions for arbitrarily large aspect ratios.

In the numerical simulations, no end conditions aretaken into account. Instead, a spanwise wavelength L is as-sumed. Hence, the adhesion points cannot be created in thesame manner as in the experiments. In fact, the adhesionmode does not seem to occur naturally from a slightly three-dimensionally perturbed 2-D solution in the whole investi-gated Reynolds number range. Yet, the vortex-adhesionmode can be induced by inserting a strong localized span-wise inhomogeneity in the initial conditions provided thatthe chosen spanwise domain L is large enough. Once thevortex adhesion mode is "excited," it is found to be self-sustaining in the range 160<Re<230. Like in the experi-ments, the numerical simulations may yield adhesion pointsfor lower Reynolds numbers 140<Re<160. While most ofthese adhesion points slowly decay, some of them seem to beself-sustaining--depending on the initial conditions. This be-havior agrees with the experimental finding that the adhesionmode can only be induced under carefully controlled condi-tions for subcritical values Re<160.

Williamson's 5 experimental finding of a hard hystereticalonset of three-dimensionality is numerically reproduced.This hard transition can be inferred from the discontinuousbehavior of the Strouhal number St, the mean drag coeffi-



0.2

0.1

0.1

0.1

0.1

0.1Cl1.1

1.

0.1

O.

C0.

0.

O.

0.

50 100 150 200 Re 300

FIG. 7. The same as Fig. 3, but including numerical data for the hardvortex-adhesion transition (0), for the soft A-mode transition (A), for thesoft C-mode transition (C), and empirical Stroubal data of Kbnig39 for hisoblique shedding mode n = 2 merging into transitional vortex shedding (*).

cient CD, and the RMS amplitude of the lift coefficient CL interms of the Reynolds number (open circles in Fig. 7). Thediscontinuity can only be observed in the presence of thevortex-adhesion mode. Otherwise, the global flow quantitiesSt, CD, and CL depend continuously on the Reynolds num-ber (open triangles in Fig. 7), like in previous numericalsimulations of Karniadakis and Triantafyllou,9 of Tomboul-ides, Triantafyllou, and Karniadakis,1 0 and of Noack andEckelmann. 11 "12 In these simulations, the chosen spanwisewavelength L did not significantly exceed three diameters. Inthis case, possible adhesion points can be, at most, a distanceL apart. Yet, our experiments and the simulations show thatadhesion points annihilate each other when they are only afew diameters apart. Upon the annihilation, the von Karmanvortices reconnect from both sides of the adhesion point andshed nearly parallel from the cylinder. No hysteretical onsetof three-dimensionality and no vortex adhesion has been nu-merically reproduced before because of the small spanwisedomain sizes in the previous simulations.

Hence, the hard, jump-like cylinder-wake transitionseems to be intimately connected with the existence of thevortex-adhesion mode. In the experiments, vortex-adhesionpoints are always induced by the end conditions for Re>160.Therefore, the hard vortex-adhesion transition appears to bethe natural onset of three-dimensionality under experimentalconditions, i.e., for cylinders with finite aspect ratio. In the

WI)CD ED SM

FIG. 8. Illustration of the A mode in the view section, 0. 5 <xlD < 1 6,O<z/D<12. Top: isovorticity surface w,= 0.02 of the numerical solutionat Re =200. Bottom: experimental flow visualization at Re= 196. The aspectratio is LID =93. The hydrogen-bubble,% wire is located at(x,,/D;y.ID) =(2; 1), i.e., only one side of the von Kirmin vortex street isvisualized.

simulations, the adhesion mode is also a self-sustaining shed-ding state, but it has to be excited by strong inhomnogeneitiesin the initial conditions.

B.2A and B made

The finite-difference computations can also reproduceWilliamson's experimental observation of two distinct span-wise patterns in the transitional Reynolds number range, hisA and B mode. According to Williamson, the A mode "rep-

resents the inception of streamwise vortex loops, for Re~180 and above" with a spanwise wavelengths around threediameters, whereupon the B mode "represents the formationof finer-scale streamline vortex pairs, for Re=230 andabove," with a spanwise length scale around one diameter(see Fig. 2 in Ref. 14).

For the computations of the A and B mode, nearly 2-Dinitial conditions were employed. The simulations were car-ried out for various spanwise domain sizes L between 6 and18 diam in order to guarantee that the simulated wake struc-tures, including the spanwise wavelengths, depend insignifi-cantly on the numerical boundary conditions.

In Fig. 8, the numerical solution and the experimentalrealization of the A mode is illustrated at Re= 200. The modedisplays a dominant spanwise wavelength of four diameters.Similarly, Williamson's B mode with a spanwise wavelengthof one diameter can be experimentally and numerically re-produced at Re=250 (see Fig. 9). The hydrogen bubbles(Fig. 9, bottom) seem to concentrate in the primary von


.90St ~~ , 0

17 - . .0 0

15 -

13

1 I I'D.6 -

4 b~O a OQAS. $++ ox Ax- s j ~~++#+44 0 EJ 600 C3

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Zhang et al.


FIG. 9. Illustration of the B mode in the view section, -0.5<x/D<16,0<z/D<18. Top: isovorticity surface &),= t 0.50 of the numerical solu-tion at Re=250. Middle: isovorticity surface co, = 0. 15 of the same solution.Note that the top and middle figures illustrate the spanwise and streamwisevorticity components, respectively. Bottom: experimental flow visualizationat Re=254, the aspect ratio being L/D=71. The hydrogen-bubble wire ispositioned at (xID;yjID)=(2;1), i.e., on one side of the von Karminvortex street.

Airman vortices and in the secondary vortices in the stream-wise direction. The primary and secondary vortices are illus-trated by numerically obtained isovorticity surfaces for thespanwise and streamwise component in Fig. 9 (top andmiddle, respectively). In the far wake, the secondary vorticeshave the tendency to merge in large-scale structures.

The spanwise structure in the near wake is displayed inFig. 10 from simulations at different Reynolds numbers. ForRe=200 and Re=240 (Fig. 10, top and bottom) the A and B

3 =

o * I A

1 Gus

0 2 4 zJD 8

FIG. 10. Spanwise structure of the A and B mode. Instantaneous contourplots of the numerically obtained vorticity component wd in the planexID 4 at Re=200, 220, and 240 (top to bottom).

mode, respectively, can be numerically reproduced in a pureform, while at intermediate Reynolds numbers around Re=220, both modes generally coexist (Fig. 10, middle).Williamson2 7 also observes a transition from the large-scaleA-mode to a fine-scale B-mode pattern around Re=230.While Williamson's wavelength of iD for the B mode iswell confirmed, our numerical and experimental value of 4Dfor the A mode is somewhat larger. The value of approxi-mately 4D is confirmed by a recent global stability analysisof Barkley and Henderson,2 9 based on a highly accuratespectral method. The numerically obtained iy-contour dia-gram for the B mode (Fig. 10, bottom) is in good qualitativeagreement with recent unpublished PiV experiments ofBrede30 for Re=400 and with the elaborate PIV study of

The A mode occurs at Re>180 and gets gradually dis-placed by the B mode at Re>230. The characteristic B-modewavelength of roughly ID can be experimentally and nu-merically observed up to at least Re= 1000. Without vortexadhesion, the A-mode transition from 2-D shedding to the3-D wake is smooth, i.e., the global flow quantities St, CD,

and CL depend continuously on the Reynolds number with asmall kink at the onset of three-dimensionality (open tri-angles in Fig. 7). In the computations, vortex adhesion at the

cylinder can be avoided by employing suitable 2-D initialconditions superimposed by a small 3-D perturbation, whichis periodic in the spanwise direction. In the experiments,localized vortex deformations are generally introduced bythe end conditions, and can be avoided by placing a thincontrol wire in the near wake or by employing other controlmechanisms.

C. C modeIn addition to the A and B mode, a different kind three-

dimensionality can be observed, called the C modde in thefollowing. This mode displays a spanwise periodicity with a

Phys. Fluids, Vol. 7, No. 4, April 1995

Z -:.N

A...n....... .... ..... .

..........

......

Zhang et al. 787


$.ft-H - t - :0

WE-j I- S=

FIG. 11. Illustration of the C mode in the view section, -0.5<x/D<16,O<z/D < 18. Top: isovorticity surface wx = 0.05 of the numerical solutionat Re=210. The thin wire is placed at (x,1D;y,1D)=(0.75;0.75). Bot-tom: experimental flow visualization at Re=215. The setup is the same as inFig. 5, except for an aspect ratio of LID =71 and a visualization wirelocated at (x,/D;y,/D)=(0.75;0.75).

wavelength of roughly 2D. It can be seen in flow visualiza-tions for 170<Re<270 (see Fig. 11, bottom) when a thinwire of diam 0.006D is placed parallel to the cylinder axisin a narrow region including the locations (xjID;yW/D)=(0;0.90) and (xID;ywID)=(0.75;0.75).2 1 Of course,the wire may also be placed symmetrically on the other sideof the plane y=O. Possibly, the wire suppresses the vortexadhesion and the A mode and impairs the B mode at 230<Re<270. Thus, the C mode can grow without being replaced bythe other 3D modes. In the present publication, no physicalmechanism for the effect of the wire can be presented.

The effect of the wire can be simulated in the finite-difference computation by setting the velocity zero on a gridline (x/D;y/D)=(0.75;0.75). The effective thickness ofthe wire is of the order of the grid size, i.e., ~-0.05D. In thiscase, also the numerical simulations yield a 3-D spanwisepattern with a wavelength of approximately 1.8D (see Fig.

_/1 -rg '

0 2 4 z/D 8

FIG. 12. Spanwise structure of the C mode. Instantaneous contour plot ofthe vorticity component xx in the plane x/D = 7.5 from the numerical simu-lation of Fig. 11.

11, top, and Fig. 12), i.e., reproduce the C mode. Experi-ments (see Fig. 45 of Ref. 21) and numerical computations(open squares in Fig. 7) yield that the C mode significantlydecreases the Strouhal number, as compared to 2-D shed-ding. This difference is larger than the corresponding fre-quency drop in the A-mode transition (open triangles in Fig.7). For Re>200, the shedding frequency is even smaller thanfor the vortex-adhesion mode (open circles in Fig. 7). Inaddition, the effect of the C mode on the lift amplitude andmean drag is large compared with the A, B, and vortex-adhesion modes. Thus, all four kinds of 3-D shedding modesreduce the St, CD, and CL values, as compared to the corre-sponding 2-D simulation. Mittal and Balachandar 3 2 confirmand explain this tendency for the lift and drag of the B modein the irregular range.

In Fig. 13, experimental values of spanwise wavelengthsare displayed in terms of the Reynolds number for differentexperimental setups. The values are seen to be discretelygrouped around one, two, and four diameters. Hence, theC-mode instability appears to be distinct from the A and Bmode.

The experimental setup with the control wire leads to asmooth C-mode transition with a 3-D periodic flow for 170<Re<200 and a quasiperiodicity with a low-frequency com-ponent for 200<Re<270. At Re>270, the flow becomes ir-regular (see Fig. 14). This behavior agrees well with thetransition scenario predicted by the low-dimensional Galer-kin model of Noack33 and Noack and Eckelmann, 11 ,12 includ-

6

X.ID

4

3

2

1

0200 300 400 Re

FIG. 13. Experimental spanwise wavelengths X, in terms of the Reynoldsnumber for various experimental setups: (0): aspect ratio L/D=71, thinwire at (x./D;y.1D)=(-1.04;0.10); ([:): L/D=71, (x,/D;yID)=(0;1.05); (A): LID=71, (x,1D;y,1D)=(0.53;0.91); (*): L/D=71,(x,.D;yID)=(0.75;0.75); (0): LID=50, (x.ID;y.ID)=(0;0.90);and (0): L/D=93, without wire.


0

3 --

Zhang et al.


0

dB-20

-40

-80 I_

dB-20

-40

-80

dB-20

-40

-60

-80

0 4 f [HZ]

FIG. 14. Experimental power spectra of the C-mode transition at Re= 177,233, and 317 (top to bottom). The cylinder diameter is 3 mm. The thin wireis placed at (xaID;yID)=(0;0.9). Thus, a dominant spanwise wavelengthof 2D is obtained (see the solid circles in Fig. 13). The spectra are obtainedfrom hot-film signals at (x/D;y/D) =(8.7;1.5).

ing the predicted wavelength of 1.8D and the critical Rey-nolds numbers of 170, 200, and 280 for three-dimensionality,quasiperiodicity, and irregularity, respectively. The quasiperi-odicity in the experimental velocity fluctuation (Fig. 14,middle) is superimposed by some background noise, whichis not present in the Galerkin results.

The most unstable 3-D Floquet mode and the asymptoticsolution of the Galerkin model display two oppositely ori-ented vortex rolls alternatingly on the upper and on the lowershear layer in the near wake (see Ref. 3). These vortex rollscan also be seen in the o),, vorticity distribution of the moreaccurate finite-difference simulations (Fig. 12). Hence, thelow-dimensional Galerkin model appears to describe thetransition scenario via the C mode. The wake resolution ofthis model seems to be too coarse for the simulation of the Aand B mode. In particular, the azimuthal resolution with a

modified trigonometric system up to only fourth order ap-pears to be too small.

In the Galerkin model, the C mode evolves naturally asthe most amplified 3-D perturbation without an imposedwake asymmetry or other control mechanisms. In the experi-ments, the C mode is excited by an asymmetric location ofthe control wire. In order to elucidate the role of this asym-metry, a symmetric setup with two wires at(x1,2/D;y1,2/D)=(0.75;+0.75) is numerically investi-gated. For this case, spanwise vortices with a slightly largerwavelength of 2.2D are observed. The slight increase of thewavelength may result from the larger effective cylinderdiam due to the displacement effect of both wires. Thus, theC mode for an effectively larger cylinder is obtained. TheC-mode structure with a spanwise wavelength of two diam-eters is also numerically obtained at Re=200 when the cyl-inder performs a transversal oscillation with a small ampli-tude and a frequency corresponding to half the naturalshedding frequency. Thus, the C mode is numerically ob-served for two different symmetric control processes. Hence,the occurrence of the C mode is not intimately connected toasymmetric excitation.

D. Comparison of the A, 13, and C mode

The vortex-adhesion mode can easily be distinguishedfrom the A, B, and C modes. The adhesion mode representsa local deformation of the primary von Kirmin vortices,whereupon the A, B, and C modes are associated with glo-bal, secondary vortices on the von Karmnn vortices. Thesethree secondary vortices are characterized by different span-wise wavelengths, but have a similar spatial structure (seeFig. 10 and Fig. 12). In this section, the A-, B-, and C-modestructures are compared in detail.

Figure 15 displays the coy vorticity component of the A-,B-, and C-mode shedding in the centerplane y = 0. The span-wise wavelength of the A and C mode are seen to character-ize the near and far wake. In contrast, the B-mode patterngoverns only the near wake. In the far wake, the one diam-eter vortices seem to merge into larger-scale structures. TheWy vorticity component of the A mode is nearly uniformlydistributed in the downstream direction. In contrast, the sec-ondary vortices of the C mode appear to concentrate in re-gions that are roughly one wavelength of the von Karmanvortex street apart. Similarly, the B-mode vortices intersectthe centerplane near the lines xID = 2, 4, and 6. Clearly, the"footprints" of the A-, B-, and C-mode vortices in the planey = 0 are distinct-apart from the wavelength. Recent PIVexperiments,3 0 and our simulations2 0 reveal that the primaryvon Kirman vortices are deformed by the secondary A-modevortices to a noticeable extent. In contrast, the von Karmanvortices are hardly deformed by the presence of the B and Cmode. Thus, Williamson 14 concludes from his flow visualiza-tions that the A-mode pattern is caused by a deformation ofthe primary vortices, whereupon the B-mode structure is dueto secondary streamwise vortices. Yet, it must be emphasizedthat also the A mode is associated with secondary vortices.This is experimentally confirmed by Fey.21



TABLE Vt. Properties and effects of the A, B, and C modes, based on thenumerical results.

A mode B mode C mode

Re range 180-230 200<Re 170-270Spanwise A z/D -4 XAD-1 XD-2

wavelengthSecondary Near and Near wake Near and

vortices far wake only far wakevon Kirmin Slightly deformed Nearly Nearlyvortex axes (4D wavelength) parallel parallelvon Kirmin Nearly Nearly Pairing in

vortex spacing equidistant equidistant x direction

FIG. 15. Secondary vortices of the A, B, and C modes (top to bottom).Instantaneous contour plots of the numerically obtained vorticity componentOy in the x,z plane at Re=200, 240, and 210 (top to bottom). The numericalsolutions for A, B, and C modes are the same as in Figs. 8, 9, and 11.

The primary von Kirmin vortices of A-, B-, and C-modeshedding are illustrated in Fig. 16. The geometry of the vor-tex street for the A and B mode are very similar. In contrast,the C-mode shedding is associated with increased down-

y/D

w1 H

0 4 8 x/D 16

FIG. 16. Primary von Kirmin vortices in the presence of the A, B, and Cmode (top to bottom). Contour plots of the numerically obtained vorticitycomponent Ao, in the x,y plane for the same instantaneous velocity fields asin Fig. 15.

stream wavelength X, of the von Karmin vortices. This in-creased X. value induces a noticeably reduced shedding fre-quency, discussed in Sec. IV C. In addition, a pairingmechanism of the von Kirmrn vortices can be seen. Thevortex pairing has also been observed in the C-mode excita-tion by a transversal cylinder oscillation (see Sec. IV C).Hence, the secondary vortices of the C mode appear to beintimately linked with the vortex pairing of the primary vonKarman vortices.

In case of the oscillating cylinder, the vortex pairing isaccompanied by a period doubling in the frequency domain.This period doubling is externally forced by a subharmonicexcitation. Interestingly, a period doubling is also observedfor a Reynolds number of Re-270 in the Galerkin model ofNoack and Eckelmann.3 The numerical simulation of Kar-niadakis and Triantafyllou 9 also yields a period doublingmechanism below Re=300. They chose a spanwise domainsize of LID=1.57, which is slightly below the C-modewavelength of XID = 1.8. The spanwise length scale of theirnear and far wake structures coincide with the domain size.Hence, their flow features appear to have more in commonwith the C mode than with the A or B mode.

The simulated control wire induces a noticeable asym-metry in the vortex shedding (Fig. 16, bottom). Yet, it shouldbe noted that the C-mode shedding is also observed in ourexperiments, in which the ratio between the control wire andthe cylinder diam is one order of magnitude smaller andtherefore the asymmetry of the boundary conditions muchless pronounced. Table VI summarizes some similarities anddifferences of the A, B, and C modes.

E. Suppression of three-dimensionality

The visualization wire placed parallel to the cylinderaxis may also be used to suppress all four 3-D instabilitymodes in a part of the transitional Reynolds number range(for details see Ref. 21). Placing the wire at(xwID;yw/D)=(1.05;0), the experimental flow visualiza-tion yields nearly 2-D periodic shedding at Re<230 (Fig. 17,bottom). In the numerical computation, 3-D fluctuations arefound to decay at Re<250, when the wire is simulated at thesame location, i.e., the stable post-transient solution repre-sents parallel shedding (Fig. 17, top). The discrepancy for theexperimentally and numerically observed onset of three-dimensionality at Re=230 and 250, respectively, may be at-tributed to the larger effective thickness of the numerical

790 Phys. Fluids, Vol. 7, No. 4, April 1995 Zhang et a/.


.

I

I.I

II

'III

IA

V

I5#

T

FIG. 17. Illustration of the suppressed onset of three-dimensionality in theview section, -0.5<xID<16, O<z/D<18. Top: isopressure surface p=-0.2 of the numerical solution at Re=220. The thin wire is placed at(x.,D;y.ID)=(1.05;0). Bottom: experimental flow visualization at Re=219. The same setup as in Fig. 5 is employed, the aspect ratio beingL/D=71. 'The visualization wire is also located at(x,,.ID;y,.1D)=(1.05;0).

wire as compared to the experimental one. The effective sup-pression of 3-D fluctuations by the wire can also be seen inthe Fourier spectra of the experimental velocity fluctuations(Fig. 18). In the range 240<Re<270, the A-, B-, and vortex-adhesion mode occur intermittently. At Re>270, the wire isfound to have little influence on the wake dynamics any-more, and the spanwise wake pattern is dominated by the Bmode.

The employed control-wire technique has been success-fully applied by Strykowski and Sreenivasan to delay theonset of periodic vortex shedding3 4 and to control theboundary-layer transition.3 5 Interestingly, the control-wirepositions, for which vortex shedding is effectively retarded,is similar to our experimentally determined region in whichthe A and B mode is suppressed, i.e., the C mode is excited.In Ref. 35, the authors emphasize that small vibrations of thecontrol wire may drastically affect the flow. In our experi-

0

dB

-20

-40

0

dB- 20

-40 .1.

- 60 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

0 2 4 6 f [Hz] 10

FIG. 18. Experimental power spectra at Re=219 without (top) and with(bottom) a wire at (x,/D;y,/D)=(1.05;0). The cylinder diameter is 4mm. The aspect ratio is L/D=71. The spectra are obtained from hot-filmsignals at (xID;yID)=(6.0;1.5).

ments, the wire displays hardly noticeable vibrations when itis located in the vicinity of (x,/D;y,/D)"(1,0). Yet, theindependent confirmation of our experimental results with asimulated stationary control-wire setup clearly shows that theretarded onset of three-dimensionality is not an artifact ofthese wire vibrations. For the C-mode setup (Sec. IV C) andvortex-adhesion setup (Sec. IV A), the wire did not vibratewithin the experimental resolution.

F. Interaction of the 3-D modes

Figure 19 summarizes the Reynolds number intervals inwhich the vortex adhesion, and the A, B, and C modes can beobserved. All four modes can be obtained in "pure" statesalong the whole cylinder span, under the conditions specifiedin the previous sections. In these shedding states, no notice-able contributions of the remaining modes are evident. Yet,there exist several overlap intervals in which these modesinteract. For instance, spanwise A- and B-mode cells, eachconsisting of several wavelengths, generally coexist in asmall Reynolds number interval around 230 according to thesimulation. In the experiments, this coexistence appears at

t 2--D mde......... -I_______

1 adhesion mode Ii__ __

1i4 i6

I A l: - - - -

I C II I

B

10 2040 220 240 260 Re

FIG. 19. Observed shedding modes and their Reynolds number ranges. Themodes are displayed in solid (dashed) boxes, when they are self-sustaining(can easily be excited under suitable conditions). For details, see the text.

Phys. Fluids, Vol. 7, No. 4, April 1995

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- :- -..

L

Zhang et a/. 791


slightly smaller Reynolds numbers around 220. For Re<230,also vortex-adhesion points perturb a neighboring spanwiseregion. This region is characterized by large shedding anglesof the von Kirmin vortices and by irregular spanwisewiggles. The vortex-adhesion points tend to retard the shed-ding of the von Kirman vortices, i.e., the shedding frequencyis reduced along the whole cylinder span. In the presence ofvortex-adhesion points, the power spectra of the experimen-tal velocity fluctuations display no peaks at a larger fre-quency corresponding to the theoretically obtained pure Amode. For suitable control-wire positions (see Sec. IV C),C-mode structures without significant A-mode or B-modecontributions may be observed in the range 170<Re<270.For Re>230, B-mode patterns with wavelengths of one di-ameter may occur intermittently in the experiments.

The above phenomenology suggests that all four modesarise from local or global 3-D instabilities of the 2-D peri-odic vortex shedding. The end conditions serve as a finitenonvanishing perturbation for the excitation of the vortex-adhesion mode, but the end conditions seem unnecessary tosustain any of the above shedding states. This conclusion isconfirmed by the fact that all 3-D modes can be numericallycomputed, assuming a periodicity in the spanwise directionand without taking end effects into account. The A, B, and Cmodes appear to result from unstable infinitesimal perturba-tions, while the vortex-adhesion mode must be excited by afinite perturbation. This hypothesis would explain why theA-mode and C-mode scenario display a soft onset of three-dimensionality, while the vortex-adhesion mode scenario ischaracterized by a hard hysteretical transition.

V. CONCLUSIONS

Williamson's 27,14 experimental observation of three dif-ferent 3-D shedding modes in the transitional cylinder wakeis numerically reproduced for the first time. These modesinclude the vortex-adhesion mode characterized by spot-likevortex deformations, a large-scale A-mode pattern in therange 180<Re<230 with a spanwise wavelength aroundfour diameters, and a fine-scale B-mode structure in therange 230<Re. In the simulation, periodic spanwise bound-ary conditions are assumed, i.e., the cylinder is effectivelyinfinitely long. Experimentally, these phenomena are ob-served for different aspect ratios and other end conditions, asemployed by Williamson. Thus, all three shedding modesseem to be able to exist as stable states independently of theend conditions. In particular, the vortex-adhesion mode isidentified as a self-sustaining shedding state. Hence, the 3-Dmodes are likely to arise from instability processes undernominally 2-D boundary conditions.

The theoretically predicted 3-D Floquet mode originat-ing at Re= 170 with a spanwise wavelength of 1.8 diameters(Noack, K6nig, and Eckelmann;' 5 Noack and Eckelmann 3) isshown to be a separate instability process, called a C mode inthis publication. For the first time, this C mode and the re-sulting spanwise structure is reproduced in the experimentsand an accurate numerical simulation, placing a thin wire atsuitable locations in the near wake. Experiments indicate a3-D wake structure with a dominant spanwise wavelength ofroughly two diameters in the range 170<Re<270.

strong 2-D thinpertur- / shedding controlbaton z wire

Re 160 7Re 18 Re 7 170

hard soft soft

|vortex-adh. | |A-mode | C-mode

'chaotic' periodic periodicI I

Re ; 200 Ret 200

'chaotic' quasi-periodic

Re 230 Re ;30 200

B3-modeA\~/D ;~ I

FIG. 20. Simplified sketch of the observed transition scenarios: the vortex-adhesion transition (left branch), the A-mode transition (middle branch), andthe C-mode transition (right branch). The retarded onset of three-dimensionality, i.e., the direct transition from 2-D shedding to the B mode atRe-230, is not included in this figure. For details, see the text.

Our present numerical and experimental results for thetransitional Reynolds number range do not suggest the exist-ence of further distinct localized or global 3-D sheddingmodes occurring naturally or under small perturbations.Even a variety of experimentally and numerically realizedstationary and periodic control processes do not give rise tothree-dimensional structures, which cannot be identified asone of the four 3-D shedding modes. For instance, the acous-tically induced "netting pattern" at Re= 143 (Fig. 9 in Ref.36) appears to be an excited A mode.

Our simulation and experiment yield four different kindsof transition scenarios: a hard vortex-adhesion, a soft Amode, a controlled C-mode, and a retarded transition (seeFig. 20). The vortex-adhesion transition with a hard hyster-etical onset of three-dimensionality can be induced numeri-cally by finite localized perturbations in the initial condi-tions, provided that the spanwise domain is large enough.This hard transition appears to be common in the experi-ments, where end effects always give rise to finite localized3-D perturbations, which originate at the cylinder ends, andare finally distributed along the whole cylinder span. TheA-mode transition with a continuous onset of three-dimensionality can be numerically obtained with nearly 2-Dinitial conditions. Both the irregular vortex-adhesion modeand the time-periodic A mode can be considered as stablecoexisting Navier-Stokes attractors roughly in the range 170<Re<230. The soft C-mode transition, predicted by the low-dimensional Galerkin model of Noack and Eckelmann, 11"12

can be numerically and experimentally observed when a thinwire is located in the near wake. Naturally, this is a con-trolled transition; the C mode appears to be too "weak" inorder to compete with the A and B mode under natural con-ditions. In the Galerkin model, the A and B modes seem tobe "suppressed" by a rather low azimuthal resolution. Fi-



nally, a retarded transition at Re=230 can be observed fromparallel shedding to the B mode by placing a control wire inthe centerplane closely behind the cylinder. This suppressionof three-dimensionality with a control-wire technique is ananalog of a similar procedure of Strkowski andSreenivasan,3 4 who suppress the onset of 2-D vortex shed-ding.

Summarizing, most present controversies on thecylinder-wake transition can easily be explained in theframework of the A, B, C, and vortex-adhesion modes. Thisclarification includes the origin of three different spanwisewavelengths of roughly one, two, and four diameters in theliterature and the role of the vortex-adhesion mode in thediscrepancy between the experimentally observed hard andthe numerically obtained soft transition. The slow irregulardynamics of the vortex-adhesion points appears also to beresponsible for the lacking experimental confirmation of thenumerically predicted 3-D time-periodic flow. The describedphenomenological aspects of the cylinder-wake transitionare believed to be of importance for many bluff-body wakes.Yet, no physical mechanisms for the four distinct 3-D insta-bilities are proposed in the present publication. Research re-sults on the physical origin of the A, B, C, and adhesionmode are the subject of a forthcoming publication.

ACKNOWLEDGMENTS

'Tvo of the authors appreciate the support of the Alex-ander von Humboldt Stiftung (H.-Q. Z.) and the DeutscheForschungsgemeinschaft Ec4l/12-2 (M. K.). We thank H.Vollmers for supporting us with his excellent visualizationsoftware "comadi" for our numerical data. We are indebtedto the rest of our former wake team at the meanwhile disin-tegrated fluid-dynamics department of the Max-Planck-Institut. In particular, we thank M. Brede and J. Wu for pro-viding us with their most recent PIV results on the spanwisestructures of the cylinder wake. We are grateful to the refer-ees for making many valuable suggestions that initiatedsome further interesting research results and that contributedto a more comprehensive presentation.

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on the transition of the cylinder wake - bernd noack

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